Problem: Factor the following expression: $-6$ $x^2$ $-25$ $x$ $-4$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-6)}{(-4)} &=& 24 \\ {a} + {b} &=& & & {-25} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $24$ and add them together. The factors that add up to ${-25}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-1}$ and ${b}$ is ${-24}$ $ \begin{eqnarray} {ab} &=& ({-1})({-24}) &=& 24 \\ {a} + {b} &=& {-1} + {-24} &=& -25 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-6}x^2 {-1}x {-24}x {-4} $ Group the terms so that there is a common factor in each group: $ ({-6}x^2 {-1}x) + ({-24}x {-4}) $ Factor out the common factors: $ x(-6x - 1) + 4(-6x - 1) $ Notice how $(-6x - 1)$ has become a common factor. Factor this out to find the answer. $(-6x - 1)(x + 4)$